How can I lock transformation of an object's vertices, edges, and faces?
In Blender, you don't need such a feature (not for preventing the edition of other objects, anyway) since Blender works in a different way. As I understand it, in 3DS Max you can select any vertex from any object at any time (unless, of course you freeze it, as you mention); however, in Blender, the procedure is to select an object and then go into Edit mode (Tab) on that object, which prevents you from editing other objects until you go back to Object mode (again, Tab).

As in this case it seems you created the second "object" while in Edit mode (which makes it a mesh, not an object), you can mouse over it and press L to it (actually, connected vertices) and P to make it into it's own object.